Integrated Symbolic Design of Unstable Nonlinear Networked Control Systems
Abstract.
The research area of Networked Control Systems (NCS) has been the topic of intensive study in the last decade. In this paper we give a contribution to this research line by addressing symbolic control design of (possibly unstable) nonlinear NCS with specifications expressed in terms of automata. We first derive symbolic models that are shown to approximate the given NCS in the sense of (alternating) approximate simulation. We then address symbolic control design with specifications expressed in terms of automata. We finally derive efficient algorithms for the synthesis of the proposed symbolic controllers that cope with the inherent computational complexity of the problem at hand.
1. Introduction
Networked Control Systems (NCS) are complex, heterogeneous, spatially distributed systems where physical processes interact with distributed computing units through non–ideal communication networks. The complexity and heterogeneity of such systems is given by the interaction of at least three components: a plant process that is often described by continuous dynamics, a controller implementing algorithms on microprocessors for the control of the plant, and a communication network conveying information between the plant and the controller which is often characterized by nonidealities such as variable sampling/transmission intervals, variable communication delays, quantization errors, packet dropouts, communication protocol and limited bandwidth. In the last decade, NCS have been the object of great interest in the research community and important results have been achieved, see e.g. [3] and the references therein. Most of the results on NCS mainly deals with stabilization problems under an imperfect communication network comprising a subset of the aforementioned communication nonidealities. The work in [1] instead, considers all the aforementioned communication nonidealities and proposes control algorithms for solving problems with complex specifications expressed in terms of automata. The main drawbacks of the results reported in [1] are:

The plant in the NCS is supposed to be stable, which is quite restrictive in many application domains of interest.

The controllers proposed require a large computational complexity in their design.
The present work improves the results established in [1] in two directions:

We extend our results to possibly unstable nonlinear networked control systems;

We design efficient algorithms that cope with the computational complexity of the approach in [1].
2. Notation
The symbols , , , , and denote the set of natural, nonnegative integer, integer, real, positive real, and nonnegative real numbers, respectively. Given a set we denote and for any . Given an interval with we denote by the set . We denote by the ceiling of a real number . Given a vector we denote by the infinity norm and by the Euclidean norm of . Given and , we set ; if then . Consider a bounded set with interior. Let be the smallest hyperrectangle containing and set . It is readily seen that for any and any there always exists such that . Given and a precision , the symbol denotes a vector in such that . Any vector with can be encoded by a finite binary word of length . Given a pair of sets and and a relation , the symbol denotes the inverse relation of , i.e. . The cardinality of a finite set is denoted by .
3. Networked Control Systems
The class of Network Control Systems (NCS) that we consider in this paper has been introduced in [1]. In this section we briefly review this model. For more details the interested reader is referred to [1]. The network scheme of the NCS is depicted in Figure 1. The direct branch of the network includes the plant , that is a nonlinear control system of the form:
(1) 
where and are the state and the control input at time , is the state space, is the set of initial states and is the set of control inputs that are supposed to be piecewise–constant functions of time from intervals of the form to . We suppose that sets and are convex, bounded and with interior. The function is such that and assumed to be Lipschitz on compact sets. In the sequel we denote by the state reached by (1) at time under the control input from the initial state ; this point is uniquely determined, since the assumptions on ensure existence and uniqueness of trajectories. We assume that the control system is forward complete, namely that every trajectory is defined on an interval of the form . On the two sides of the plant in Figure 1, a ZeroorderHolder (ZoH) and a (ideal) sensor are placed. We assume that the ZoH and the sensor are synchronized and update their output values at times that are integer multiples of the same interval , i.e. , , , , where is the index of the sampling interval (starting from ). The evolution of the NCS is described iteratively in the following, starting from the initial time . Consider the –th iteration in the feedback loop. The sensor requests access to the network and after a waiting time , it sends at time the latest available sample where is the precision of the quantizer that follows the sensor in the NCS scheme in Figure 1. The sensortocontroller (sc) link of the network introduces a delay , with , where is the minimum time required to send the information over the sensortocontroller branch, assuming a digital communication channel of bandwitdh (expressed in bits per second (bps)). The maximum network delay takes into account congestion, other accesses to the communication channel, any kind of scheduling protocol and a finite number of subsequent packet dropouts, which is assumed to be uniformly bounded. After that time, the sensor sample reaches the symbolic controller, that is expressed in terms of the function , with and so that the domain and co–domain of are non–empty. After a time , the value is returned and it is sent through the network at time (after a bounded waiting time ). The controllertoactuator (ca) link of the network introduces a delay , where and is the minimum time required to send the information over the controllertoactuator branch of the network. After that time, the sample reaches the ZoH and at time the ZoH is refreshed to the control value , with . The next iteration starts and the sensor requests access to the network again. Consider now the sequence of control values . Each value is held for sampling intervals. Due to the bounded delays, one gets , with:
(2) 
where we set , . In the sequel we refer to the described NCS by and to a trajectory of with initial state and control input by .
4. Systems, Approximate Equivalence and Composition
We use the notion of system as a unified mathematical framework to describe NCS as well as their symbolic models.
Definition 4.1.
[6] A system is a sextuple consisting of:

a set of states ;

a set of initial states ;

a set of inputs ;

a transition relation ;

a set of outputs ;

an output function .
A transition is denoted by . For such a transition, state is called a successor, or simply a successor, of state .
A state run of is a (possibly infinite) sequence of transitions with . An output run is a (possibly infinite) sequence such that there exists a state run with , . System is said to be:

countable if and are countable sets;

symbolic if and are finite sets;

metric if the output set is equipped with a metric ;

deterministic if for any and there exists at most one state such that for some ;

non–blocking if for any there exists at least one state such that for some ;

accessible, if for any there exists a finite number of transitions from an initial state to state .
Definition 4.2.
Given two systems (), is a sub–system of , denoted , if , , , , , and for any .
In the sequel we consider (alternating) approximate simulation relations [6] to relate properties of NCS and symbolic models.
Definition 4.3.
[2, 5] Let () be metric systems with the same output sets and metric , and let be a given precision. Consider a relation satisfying the following conditions:

such that ;

, .
Relation is an –approximate simulation relation from to if it enjoys conditions (i), (ii) and the following one:

such that .
System is –simulated by or –simulates , denoted , if there exists an –approximate simulation relation from to . Relation is an alternating –approximate () simulation relation from to if it enjoys conditions (i), (ii) and the following one:

such that .
System is alternating –simulated by or alternating –simulates , denoted , if there exists an simulation relation from to .
For more details on the above notions we refer to [6, 2, 5]. We conclude this section with the notion of approximate feedback composition, that is employed in the sequel to capture feedback interaction between nondeterministic systems and symbolic controllers.
Definition 4.4.
[6] Consider a pair of metric systems () with the same output sets and metric . Let be an simulation relation from to . The –approximate feedback composition of and , with composition relation , is the system , where

;

;

;

if and ;

;

for any .
5. Symbolic Models for NCS
In this section we propose symbolic models that approximate NCS in the sense of (alternating) approximate simulation. For notational simplicity we denote by any constant control input s.t. at all times . Set .
Definition 5.1.

is the subset of such that for any , with , the following conditions hold:
(3) (4) for some constant functions , ;

;

;

, where
for some ;

;

.
Note that is nondeterministic because, depending on the values of , more than one –successor of may exist. Since the state vectors of are built from trajectories of sampled every time units, collects all the information of the NCS available at the sensor (see Figure 1) as formally stated in Theorem 5.1 of [1]. System can be regarded as metric with the metric on naturally induced by the metric on , as follows. Given any , , we set , if and , otherwise. Although system contains all the information of the NCS available at the sensor, it is not a finite model. In the following, we propose a system that approximates and is symbolic. A key property for our developments is the notion of incremental forward completeness, as recalled hereafter.
Definition 5.2.
Incremental forward completeness requires the distance between two arbitrary trajectories to be bounded by a continuous function capturing the mismatch between initial conditions. The class of FC control systems is rather large and includes also some subclasses of unstable control systems; for instance unstable linear systems are FC. The notion of FC can be described in terms of Lyapunovlike functions.
Definition 5.3.
A smooth function is called a –FC Lyapunov function for the control system (1) if there exist and functions and such that, for any and any , the following conditions hold true:

,

.
The existence of a FC Lyapunov function was proven in [7] to be a sufficient condition for FC of a control system. In the following we suppose that the control system in the NCS enjoys the following properties:

There exists a –FC Lyapunov function satisfying the inequality (ii) in Definition 5.3 for some ;

There exists a function such that , for every .
Given a design parameter , define the following system
where:

is the subset of such that for any with the following condition holds:
(5) (6) for some constant functions , ;

;

;

, where
for some ;

;

.
System is metric when we regard the set of outputs as being equipped with the metric . We now have all the ingredients to present one of the main results of this paper.
Theorem 5.4.
Consider the NCS and suppose that the control system enjoys properties (H1) and (H2). Then for any desired precision , any sampling time , any state quantization and any choice of the design parameter satisfying the inequality
(7) 
we have .
Proof.
First we prove that , according to Definition 4.3. Consider the relation defined by if and only if:

, , for some ;

for ;
Conditions (i) and (ii) in Definition 4.3 can be proven by using similar arguments employed in the proof of Theorem 5.8 in [1]. We now show that condition (iii) in Definition 4.3 holds. Consider any , with , , for some , and any ; then pick and consider any transition , with , for some . Pick defined by for all . We now prove that is a transition of . First, from condition (i) in Definition 5.3, the definition of and the first inequality in (7), one can write:
(8) 
for all . By Assumption (H1), condition (ii) in Definition 5.3 writes:
(9) 
By considering Assumption (H2), the definitions of and , and by integrating the previous inequality, the following holds:
(10)  
where condition in (7) has been used in the last step. By similar computations, it is possible to prove that the inequality in (8) implies:
(11)  
(12) 
Hence, from the inequalities in (10)–(12) and from the definition of the transition relation in , the transition is in , implying with (8) that , which concludes the proof of condition (iii) of Definition 4.3. We now prove , according to Definition 4.3, by considering the relation . We prove condition (iii) in Definition 4.3, because the proof of condition (i) is given in [1], while condition (ii) is fulfilled for the relation because it has been proved to hold for . Consider any , with , , for some , and any transition in , for some , with for some . Pick defined by for all . By using similar arguments as in the proof of condition (iii) of Definition 4.3 for the relation , it is possible to show that the transition , with , is in , and that for all , hence , which concludes the proof. ∎
This result is important because it provides symbolic models for possibly unstable nonlinear NCS, with guaranteed approximation bounds. This result generalizes the ones in [1], which instead require incrementally stable NCS.
6. Robust symbolic Control Design
We consider a control design problem where the NCS has to satisfy a given specification robustly with respect to the nonidealities of the communication network. Our specification is a collection of transitions , where is a finite subset of . Given a set of initial states , we now reformulate the specification in the form of the system
where:

is the subset of such that for any , with , for any , the transition is in ;

;

, where is a dummy symbol;

, where , , and the transition is in ;

;

,
where and are as in (2). We are now ready to state the control problem that we address in this section.
Problem 6.1.
Consider the NCS , a specification and a desired precision . Find a symbolic controller , a parameter and a simulation relation from to such that:

;

is nonblocking.
Note that the approximate similarity inclusion in (1) requires the state trajectories of the NCS to be close to the ones of specification up to the accuracy robustly with respect to the nondeterminism imposed by the network. The nonblocking condition (2) prevents deadlocks in the interaction between the plant and the controller. In the following definition, we provide the controller that is shown in the sequel to solve Problem 6.1.
Definition 6.2.
Let be the maximal nonblocking subsystem
From the above definition it is easy to see that is symbolic. The following technical result will be useful in the sequel.
Lemma 6.3.
Let (, , ) be metric systems with the same output sets and metric . Then the following statements hold: